[[176,6,19]] · QEC Challenge

[[176,6,19]] d ≤

176

kd²/n

12.307

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 19, d_Z ≤ 19 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 19 · witness weight 19 (claimed upper_bound)

witness operator (support, 19 qubits)

[1, 2, 21, 27, 35, 56, 60, 62, 64, 66, 80, 93, 96, 100, 106, 123, 135, 152, 169]

d_Z 19 · witness weight 19 (claimed upper_bound)

witness operator (support, 19 qubits)

[20, 22, 33, 37, 40, 45, 52, 67, 69, 71, 89, 112, 115, 132, 141, 150, 157, 159, 167]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking

girth H_X 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×176 (2,4)×88 (3,4)×88 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,4): 176 (2,4): 88 (2,6): 2288 (3,4): 88 (3,6): 3872 (3,8): 40568 (3,10): 3960

trapping sets H_Z (1,4)×176 (2,4)×88 (3,4)×88 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,4): 176 (2,4): 88 (2,6): 2288 (3,4): 88 (3,6): 3872 (3,8): 40568 (3,10): 3960

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(88,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order88_k6.txt, line 7. nonidentity supports a=[2, 3, 14], b=[5, 33, 58].

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-19

notes Literature reproduction of a published 2BGA code (arXiv:2306.16400; github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c). Checked against the live board: not equivalent to any existing entry.

family generalized bicycle (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[176,6,19]] — two-block group-algebra code on SmallGroup(88,1)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 12.31.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts and parameter tables. From github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order88_k6.txt, line 7. This row is SmallGroup(88,1) with nonidentity GAP supports a=[2, 3, 14], b=[5, 33, 58]. Reconstructed via GAP (Cayley table) and the repository's group-algebra constructor.

Evidence trail

The trusted candidate gate returned passed: true, structural verification passed (CSS, connectivity, k=6), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 19, not an exact certificate. The published distance is consistent with this run.

Dead ends

None for this candidate. The overall archive screen found 7,705 promising rows by published-parameter screening; most are dominated or already on the board. This row survived the screen and the gate.

Tools

DeepSeek V4 Flash 0731 (provenance.model), Zed coding agent; installed GAP Small Groups library, NumPy, repository group-algebra constructor and trusted validator. Public artifacts fetched by immutable commit URL and treated as data. Bounded local CPU run, no paid compute.

Reproduction

Use GAP g := SmallGroup(88,1), e := Elements(g). Construct the zero-based Cayley table mul[i,j] = Position(e,e[i+1]*e[j+1])-1. The published nonidentity supports are a=[2, 3, 14], b=[5, 33, 58] in GAP's one-based indexing. Add identity and convert to zero-based. Build HX=[L(a)|R(b)], HZ=[R(b)^T|L(a)^T] with left/right regular representations. Verify CSS; k = 176 - rank(HX) - rank(HZ) = 6. Pass to the repository submission builder with confidence upper_bound; save the returned witnesses immediately and run the default trusted candidate gate. Do not replace the returned distance with the paper's number.

Parity checks

X-checks 88 (max weight 8) · Z-checks 88 (max weight 8)

H_X (88 checks, sparse supports)

[0, 8, 12, 14, 88, 128, 153, 163] [0, 1, 12, 86, 89, 132, 157, 167] [0, 1, 2, 20, 90, 115, 154, 169] [2, 3, 5, 7, 91, 121, 160, 170] [4, 16, 20, 22, 88, 92, 136, 161] [1, 2, 5, 75, 93, 119, 158, 172] [3, 5, 6, 81, 94, 125, 164, 173] [4, 7, 8, 20, 89, 95, 140, 165] [3, 6, 8, 13, 96, 122, 147, 174] [4, 7, 9, 28, 90, 97, 123, 162] [9, 10, 13, 15, 91, 98, 129, 168] [11, 24, 28, 30, 92, 99, 144, 169] [6, 8, 12, 82, 100, 126, 151, 175] [0, 7, 9, 13, 93, 101, 127, 166] [2, 10, 13, 14, 94, 102, 133, 171] [11, 15, 16, 28, 95, 103, 148, 172] [10, 14, 16, 21, 96, 104, 130, 155] [11, 15, 17, 36, 97, 105, 131, 170] [17, 18, 21, 23, 98, 106, 137, 174] [19, 32, 36, 38, 90, 99, 107, 152] [3, 14, 16, 20, 100, 108, 134, 159] [4, 15, 17, 21, 101, 109, 135, 173] [9, 18, 21, 22, 102, 110, 141, 175] [19, 23, 24, 36, 93, 103, 111, 156] [18, 22, 24, 29, 104, 112, 138, 163] [19, 23, 25, 44, 91, 105, 113, 139] [25, 26, 29, 31, 96, 106, 114, 145] [27, 40, 44, 46, 97, 107, 115, 160] [10, 22, 24, 28, 108, 116, 142, 167] [11, 23, 25, 29, 94, 109, 117, 143] [17, 26, 29, 30, 100, 110, 118, 149] [27, 31, 32, 44, 101, 111, 119, 164] [26, 30, 32, 37, 88, 112, 120, 146] [27, 31, 33, 52, 98, 113, 121, 147] [33, 34, 37, 39, 104, 114, 122, 153] [35, 48, 52, 54, 105, 115, 123, 168] [18, 30, 32, 36, 89, 116, 124, 150] [19, 31, 33, 37, 102, 117, 125, 151] [25, 34, 37, 38, 108, 118, 126, 157] [35, 39, 40, 52, 109, 119, 127, 171] [34, 38, 40, 45, 92, 120, 128, 154] [35, 39, 41, 60, 106, 121, 129, 155] [41, 42, 45, 47, 112, 122, 130, 161] [43, 56, 60, 62, 113, 123, 131, 174] [26, 38, 40, 44, 95, 124, 132, 158] [27, 39, 41, 45, 110, 125, 133, 159] [33, 42, 45, 46, 116, 126, 134, 165] [43, 47, 48, 60, 117, 127, 135, 175] [42, 46, 48, 53, 99, 128, 136, 162] [43, 47, 49, 68, 114, 129, 137, 163] [49, 50, 53, 55, 120, 130, 138, 169] [51, 64, 68, 70, 96, 121, 131, 139] [34, 46, 48, 52, 103, 132, 140, 166] [35, 47, 49, 53, 118, 133, 141, 167] [41, 50, 53, 54, 124, 134, 142, 172] [51, 55, 56, 68, 100, 125, 135, 143] [50, 54, 56, 61, 107, 136, 144, 170] [51, 55, 57, 76, 88, 122, 137, 145] [57, 58, 61, 63, 90, 128, 138, 146] [59, 72, 76, 78, 104, 129, 139, 147] [42, 54, 56, 60, 111, 140, 148, 173] [43, 55, 57, 61, 89, 126, 141, 149] [49, 58, 61, 62, 93, 132, 142, 150] [59, 63, 64, 76, 108, 133, 143, 151] [58, 62, 64, 69, 91, 115, 144, 152] [59, 63, 65, 83, 92, 130, 145, 153] [65, 66, 69, 71, 97, 136, 146, 154] [67, 80, 83, 85, 112, 137, 147, 155] [50, 62, 64, 68, 94, 119, 148, 156] [51, 63, 65, 69, 95, 134, 149, 157] [57, 66, 69, 70, 101, 140, 150, 158] [67, 71, 72, 83, 116, 141, 151, 159] [66, 70, 72, 77, 98, 123, 152, 160] [67, 71, 73, 87, 99, 138, 153, 161] [73, 74, 77, 79, 105, 144, 154, 162] [6, 75, 86, 87, 120, 145, 155, 163] [58, 70, 72, 76, 102, 127, 156, 164] [59, 71, 73, 77, 103, 142, 157, 165] [65, 74, 77, 78, 109, 148, 158, 166] [75, 79, 80, 87, 124, 149, 159, 167] [74, 78, 80, 84, 106, 131, 160, 168] [12, 75, 79, 81, 107, 146, 161, 169] [1, 81, 82, 84, 113, 152, 162, 170] [66, 78, 80, 83, 110, 135, 164, 171] [67, 79, 81, 84, 111, 150, 165, 172] [73, 82, 84, 85, 117, 156, 166, 173] [5, 82, 85, 86, 114, 139, 168, 174] [74, 85, 86, 87, 118, 143, 171, 175]

H_Z (88 checks, sparse supports)

[0, 4, 32, 57, 88, 89, 90, 101] [1, 7, 36, 61, 89, 90, 93, 170] [2, 9, 19, 58, 90, 91, 93, 102] [3, 10, 25, 64, 91, 94, 96, 108] [4, 11, 40, 65, 92, 95, 97, 109] [5, 13, 23, 62, 91, 93, 94, 174] [6, 14, 29, 68, 94, 96, 100, 163] [7, 15, 44, 69, 91, 95, 97, 101] [8, 16, 26, 51, 88, 95, 96, 100] [9, 17, 27, 66, 97, 98, 101, 110] [10, 18, 33, 72, 98, 102, 104, 116] [11, 19, 48, 73, 99, 103, 105, 117] [12, 20, 30, 55, 88, 89, 100, 169] [13, 21, 31, 70, 96, 98, 101, 102] [14, 22, 37, 76, 88, 102, 104, 108] [15, 23, 52, 77, 98, 103, 105, 109] [16, 24, 34, 59, 92, 103, 104, 108] [17, 25, 35, 74, 105, 106, 109, 118] [18, 26, 41, 80, 106, 110, 112, 124] [19, 27, 56, 81, 107, 111, 113, 125] [20, 28, 38, 63, 90, 92, 95, 108] [21, 29, 39, 78, 104, 106, 109, 110] [22, 30, 45, 83, 92, 110, 112, 116] [23, 31, 60, 84, 106, 111, 113, 117] [24, 32, 42, 67, 99, 111, 112, 116] [25, 33, 43, 82, 113, 114, 117, 126] [26, 34, 49, 86, 114, 118, 120, 132] [2, 27, 35, 64, 115, 119, 121, 133] [28, 36, 46, 71, 97, 99, 103, 116] [29, 37, 47, 85, 112, 114, 117, 118] [30, 38, 53, 87, 99, 118, 120, 124] [5, 31, 39, 68, 114, 119, 121, 125] [32, 40, 50, 75, 107, 119, 120, 124] [3, 33, 41, 51, 121, 122, 125, 134] [8, 34, 42, 57, 122, 126, 128, 140] [9, 35, 43, 72, 123, 127, 129, 141] [36, 44, 54, 79, 105, 107, 111, 124] [6, 37, 45, 55, 120, 122, 125, 126] [12, 38, 46, 61, 107, 126, 128, 132] [13, 39, 47, 76, 122, 127, 129, 133] [0, 40, 48, 58, 115, 127, 128, 132] [10, 41, 49, 59, 129, 130, 133, 142] [16, 42, 50, 65, 130, 134, 136, 148] [17, 43, 51, 80, 131, 135, 137, 149] [1, 44, 52, 62, 113, 115, 119, 132] [14, 45, 53, 63, 128, 130, 133, 134] [20, 46, 54, 69, 115, 134, 136, 140] [21, 47, 55, 83, 130, 135, 137, 141] [4, 48, 56, 66, 123, 135, 136, 140] [18, 49, 57, 67, 137, 138, 141, 150] [24, 50, 58, 73, 138, 142, 144, 156] [25, 51, 59, 86, 139, 143, 145, 157] [7, 52, 60, 70, 121, 123, 127, 140] [22, 53, 61, 71, 136, 138, 141, 142] [28, 54, 62, 77, 123, 142, 144, 148] [29, 55, 63, 87, 138, 143, 145, 149] [11, 56, 64, 74, 131, 143, 144, 148] [26, 57, 65, 75, 145, 146, 149, 158] [32, 58, 66, 81, 146, 150, 152, 164] [8, 33, 59, 67, 147, 151, 153, 165] [15, 60, 68, 78, 129, 131, 135, 148] [30, 61, 69, 79, 144, 146, 149, 150] [36, 62, 70, 84, 131, 150, 152, 156] [12, 37, 63, 71, 146, 151, 153, 157] [19, 64, 72, 82, 139, 151, 152, 156] [0, 34, 65, 73, 153, 154, 157, 166] [2, 40, 66, 74, 154, 158, 160, 171] [16, 41, 67, 75, 155, 159, 161, 172] [23, 68, 76, 85, 137, 139, 143, 156] [1, 38, 69, 77, 152, 154, 157, 158] [5, 44, 70, 78, 139, 158, 160, 164] [20, 45, 71, 79, 154, 159, 161, 165] [3, 27, 72, 80, 147, 159, 160, 164] [4, 42, 73, 81, 161, 162, 165, 173] [9, 48, 74, 82, 162, 166, 168, 175] [0, 24, 49, 75, 93, 163, 167, 169] [6, 31, 76, 83, 145, 147, 151, 164] [7, 46, 77, 84, 160, 162, 165, 166] [13, 52, 78, 85, 147, 166, 168, 171] [1, 28, 53, 79, 162, 167, 169, 172] [10, 35, 80, 86, 155, 167, 168, 171] [2, 11, 50, 81, 94, 169, 170, 172] [3, 17, 56, 82, 100, 170, 173, 174] [14, 39, 83, 87, 153, 155, 159, 171] [5, 15, 54, 84, 168, 170, 172, 173] [6, 21, 60, 85, 155, 173, 174, 175] [8, 18, 43, 86, 89, 163, 174, 175] [12, 22, 47, 87, 161, 163, 167, 175]

Code ID 176-6-19 · download JSON · raw on GitHub